Spiral Curves

In engineering construction, the surveyor often inserts a transition curve, also known as a spiral curve, between a circular curve and the tangent to that curve. The spiral is a curve of varying radius used to gradually increase the curvature of a road or railroad. Spiral curves are used primarily to reduce skidding and steering difficulties by gradual transition between straight-line and turning motion, and/or to provide a method for adequately superelevating curves.

The spiral curve is designed to provide for a gradual superelevation of the outer pavement edge of the road to counteract the centrifugal force of vehicles as they pass. The best spiral curve is one in which the superelevation increases uniformly with the length of the spiral from the TS or the point where the spiral curve leaves the tangent.

The curvature of a spiral must increase uniformly from its beginning to its end. At the beginning, where it leaves the tangent, its curvature is zero; at the end, where it joins the circular curve, it has the same degree of curvature as the circular curve it intercepts.

Theory of A.R.E.A. 10-Chord Spiral
The spiral of the American Railway Engineering Association, known as the A.R.E.A. spiral, retains nearly all the characteristics of the cubic spiral. In the cubic spiral, the lengths have been considered as measured along the spiral curve itself, but measurements in the field must be taken by chords. Recognizing this fact, in the A.R.E.A. spiral the length of spiral is measured by 10 equal chords, so that the theoretical curve is brought into harmony with field practice. This 10-chord spiral closely approximates the cubic spiral. Basically, the two curves coincide up to the point where Δ = 15 degrees. The exact formulas for this A.R.E.A. 10-chord spiral, when Δ does not exeed 45 degrees, are given on pages 28 and 29.

Spiral Elements
Figures 15 and 16 show the notations applied to elements of a simple circular curve with spirals connecting it to the tangents.

TS = the point of change from tangent to spiral
SC = the point of change from spiral to circular curve
CS = the point of change from circular curve to spiral
ST = the point of change from spiral to tangent
SS = the point of change from one spiral to another
(not shown in figure 15 or figure 16)

The symbols PC and PT, TS and ST, and SC and CS become transposed when the direction of stationing is changed.

a = the angle between the tangent at the TS and the chord from the TS to any point on the spiral
A = the angle between the tangent at the TS and the chord from the TS to the SC
b = the angle at any point on the spiral between the tangent at that point and the chord from the TS
B = the angle at the SC between the chord from the TS and the tangent at the SC
c = the chord from any point on the spiral to the TS
C = the chord from the TS to the SC
d = the degree of curve at any point on the spiral
D = the degree of curve of the circular arc
f = the angle between any chord of the spiral (calculated when necessary) and the tangent through the TS
I = the angle of the deflection between initial and final tangents; the total central angle of the circular curve and spirals
k = the increase in degree of curve per station on the spiral
L = the length of the spiral in feet from the TS to any given point on the spiral
L_s = the length of the spiral in feet from the TS to the SC, measured in 10 equal chords
o = the ordinate of the offsetted PC; the distance between the tangent and a parallel tangent to the offsetted curve
r = the radius of the osculating circle at any given point of the spiral
R = the radius of the central circular curve
s = the length of the spiral in stations from the TS to any given point
S = the length of the spiral in stations from the TS to the SC
u = the distance on the tangent from the TS to the intersection with a tangent through any given point on the spiral
U = the distance on the tangent from the TS to the intersection with a tangent through the SC; the longer spiral tangent
v = the distance on the tangent through any given point from that point to the intersection with the tangent through the TS
V = the distance on the tangent through the SC from the SC to the intersection with the tangent through the TS; the shorter spiral tangent
x = the tangent distance from the TS to any point on the spiral
X = the tangent distance from the TS to the SC
y = the tangent offset of any point on the spiral
Y = the tangent offset of the SC
Z = the tangent distance from the TS to the offsetted PC (Z = X/2, approximately)
δ = the central angle of the spiral from the TS to any given point 
Δ = the central angle of the whole spiral
T_s = the tangent distance of the spiraled curve; distance from TS to PI, the point of intersection of tangents
E_s = the external distance of the offsetted curve

Spiral Formulas
The following formulas are for the exact determination of the functions of the 10- chord spiral when the central angle, Δ, does not exceed 45 degrees. These are suitable for the compilation of tables and for accurate fieldwork.

1. d = ks = kL / 100
2. D = kS = kL_s / 100
3. δ = ks² / 2 = d_s / 2 = kL² / 20,000 = DL / 200
4. ks² / 2 = DS / 2 = kL_s / 20,000 = DL_s / 200
5. A = (Δ/3) – 0.00297 Δ³ seconds
6. B = Δ – A
7. C = L_s (Cos 0.3 Δ + 0.004 Exsec ¾ Δ)
   (Exsec Δ = 1 Tan ½ (Δ)
8. X = C Cos A
9. Y = C Sin A
10. U = C (Sin B / Sin Δ)
11. V = C (Sin A / Sin Δ)
12. R = 50 ft / Sin ½ D (chord definition)
13. Z = X – (R Sin Δ)
14. o = Y – (R Vers Δ)
    (Vers Δ = 1 – Cos Δ)
15. T_s = (R + o) Tan (½ I) + Z
16. Es = (R + o) Exsec (½ I) +o
    (Exsec (½ I) = Tan (½ I) (Tan (¼ I))

Empirical Formulas
For use in the field, the following formulas are sufficiently accurate for practical purposes when Δ does not exceed 15 degrees.
a = δ/3 (degrees)
A = Δ/3 (degrees)
a = 10 ks2 (minutes)
S = 10 kS2(minutes)

Spiral Lengths
Different factors must be taken into account when calculating spiral lengths for highway and railroad layout.

Highways. Spirals applied to highway layout must be long enough to permit the effects of centrifugal force to be adequately compensated for by proper superelevation. The minimum transition spiral length for any degree of curvature and design speed is obtained from the the relationship L_s= 1.6V3/R, in which Ls is the minimum spiral length in feet, V is the design speed in miles per hour, and R is the radius of curvature of the simple curve. This equation is not mathematically exact but an approximation based on years of observation and road tests.

Table 1 is compiled from the above equation for multiples of 50 feet. When spirals are inserted between the arcs of a compound curve, use L_s= 1.6V3/Ra. Ra represents the radius of a curve of a degree equal to the difference in degrees of curvature of the circular arcs.

Railroads Spirals applied to railroad layout must be long enough to permit an increase in superelevation not exceeding 1¼ inches per second for the maximum speed of train operation. The minimum length is determined from the equation L_s= 1.17 EV. E is the full theoretical superelevation of the curve in inches, V is the speed in miles per hour, and L_s is the spiral length in feet.

This length of spiral provides the best riding conditions by maintaining the desired relationship between the amount of superelevation and the degree of curvature. The degree of curvature increases uniformly throughout the length of the spiral. The same equation is used to compute the length of a spiral between the arcs of a compound curve. In such a case, E is the difference between the superelevations of the two circular arcs.